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# orick2 - and b = 3 s t 7(12 pts Find the directional...

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MATH 22 Exam 2 Name: ______________________ FALL, 2000 1. (10 pts.) Find the domain and range of the function f ( x , y , z ) = z 2 ln(4 x + 2 y - z ) . 2. (10 pts.) Show that the limit lim ( x , y ) (0,0) xy 2 x 2 + y 4 does not exist by considering the limiting values along the two paths x = 0 and x = y 2 . 3. (8 pts.) Determine where the following function is continuous and explain your reasoning g ( x , y ) = xy 2 x 2 + y 4 ( x , y ) (0,0) 0 ( x , y ) = (0,0) 4. (16 pts.) Given u ( x , y ) = e - x cos( y ) - e - y cos( x ), find the second partials of u and show that u is a solution to Laplace’s equation: u xx + u yy = 0. 5. (12 pts.) Find the equation of the tangent plane and normal line to the surface 9 x 2 - 4 y 2 + z 2 = 10 at the point (1, 0, -1). 6. (8 pts.) Use the chain rule to find w t where w = sin( a )cos( b ) , with a = st 2
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Unformatted text preview: , and b = 3 s + t . 7. (12 pts.) Find the directional derivative of f ( x , y ) = 4 x 2-y 2 at the point (1,1) in the direction of v = 4,-1 . Is this the direction of maximum rate of change at the point (1,1)? YES NO (Circle one) (12 pts.) Find all local extreme values and/or saddle points of the function f ( x , y ) = e-x cos( y ). (12 pts.) Set up the system of 5 equations necessary to apply the method of Lagrange Multipliers to find the maximum and minimum values of f ( x , y , z ) = x 2-y 2 + z 2 subject to the two constraints g ( x , y , z ) = 3 x-2 y + z = 4 h ( x , y , z ) = x 2 + y 2 = 4 It is not necessary to solve this system of equations....
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